Question

Pranav invests $3,313 in a retirement account with a fixed annual interest rate compounded continuously. After 20 years, the balance reaches $7,373.22. What is the interest rate of the account.

Answer 1

Interest rate of the account is 4%

Step-by-step explanation:

initial amount = $3313

time = 20 years

Balance = future value = $7373.22

rate = ?

n = number of times compounded

n = continuously

The formula for continuous compounding:

`P=P_0e^(rt)`
`\begin{gathered} P\text{ = amount after a certain time }=\text{ \$7373.22} \\ P_0\text{ = \$3313, t = 20 , r = ?} \\ \text{Substitute in the formula:} \\ \text{7373.22 = 3313e}^(r*20) \\ \text{7373.22 = 3313e}^(20r) \\ \\ \text{divide both sides by 3313:} \\ \frac{\text{7373.22}}{3313}\text{ = }\frac{\text{3313e}^(20r)}{3313} \end{gathered}`
`\begin{gathered} \frac{\text{7373.22}}{3313}\text{ = e}^(20r) \\ 2.2255\text{ = e}^(20r) \\ \text{Taking natural log of both sides:} \\ \ln \text{ 2.2255 = ln}(\text{e}^(20r)) \\ 0.8\text{ = 20r} \end{gathered}`
`\begin{gathered} \text{divide both sides by 20:} \\ (0.8)/(20)\text{ = }(20r)/(20) \\ r\text{ = 0.04} \\ \\ In\text{ interest rate in percent = 0.04 }*\text{ 100\% } \\ \text{interest rate in percent = 4\%} \end{gathered}`

Interest rate of the account is 4%